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Due to the fermion sign problem, standard lattice Monte-Carlo method for QCD fail at small temperatures and high baryon densities. $G_2$-QCD, QCD with the gauge group $SU(3)$ replaced by the exceptional Lie group $G_2$, can be simulated using lattice techniques at these densities, and can therefore provide an illustration of the possible phase structure. Here we present a systematic investigation of the ground-state hadronic spectrum using lattice simulations for different quark masses in several hadronic sectors. We then show that the different hadronic scales of Goldstone bosons, intermediate bosons, and baryons is reflected in the phase structure at finite density.

G2-QCD: Spectroscopy and the phase diagram at zero temperature and finite density